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A-Level CAIE Mathematics A21.2 FunctionsQuestion Bank

1.2 Functions

Question 1

Question 1(b)

(a)

The curve with equation y=x2y=x^{2} is transformed to the curve with equation y=x2+6x+5y=x^{2}+6 x+5.

Describe fully the transformation(s) involved.

[ 2 ]

Question 1

Functions f and g are defined by

f:x↦10−3x,x∈R, g:x↦103−2x,x∈R,x≠32\begin{aligned} & \mathrm{f}: x \mapsto 10-3 x, \quad x \in \mathbb{R}, \\ & \mathrm{~g}: x \mapsto \frac{10}{3-2 x}, \quad x \in \mathbb{R}, x \neq \frac{3}{2} \end{aligned}

Solve the equation ff(x)=gf(2).

Question 1

[Maximum number: 4]

Functions f and g are defined by

f:x↦3x+2,x∈R, g:x↦4x−12,x∈R.\begin{aligned} & \mathrm{f}: x \mapsto 3 x+2, \quad x \in \mathbb{R}, \\ & \mathrm{~g}: x \mapsto 4 x-12, \quad x \in \mathbb{R} . \end{aligned}

Solve the equation f−1(x)=gf(x)\mathrm{f}^{-1}(x)=\mathrm{gf}(x).

Question 1

[Maximum number: 4]

The graph of y=f(x) is transformed to the graph of y=3-f(x).
Describe fully, in the correct order, the two transformations that have been combined.

Question 2

[Maximum number: 4]

The graph of y=f(x) is transformed to the graph of y=1+f(12x)y=1+\mathrm{f}\left(\frac{1}{2} x\right).
Describe fully the two single transformations which have been combined to give the resulting transformation.

Question 2

[Maximum number: 6]

A function f is defined by f : x↦4−5xx \mapsto 4-5 x for x∈Rx \in \mathbb{R}.

Question 2(i)

(a)

Find an expression for f−1(x)\mathrm{f}^{-1}(x) and find the point of intersection of the graphs of y=f(x) and y=f−1(x)y=\mathrm{f}^{-1}(x).

[ 3 ]

Question 2(ii)

(b)

Sketch, on the same diagram, the graphs of y=f(x) and y=f−1(x)y=\mathrm{f}^{-1}(x), making clear the relationship between the graphs.

[ 3 ]

Question 2

[Maximum number: 4]

The function f is defined by f(x)=−2x2−8x−13\mathrm{f}(x)=-2 x^{2}-8 x-13 for x<-3.

Question 2(b)

(a)

Find the range of f.

[ 1 ]

Question 2(c)

(b)

Find an expression for f−1(x)\mathrm{f}^{-1}(x).

[ 3 ]

Question 2

[Maximum number: 4]

A function f is such that f(x)=(x+32)+1\mathrm{f}(x)=\sqrt{ }\left(\frac{x+3}{2}\right)+1, for x⩾−3x \geqslant-3. Find

Question 2(i)

(a)

f−1(x)\mathrm{f}^{-1}(x) in the form ax2+bx+ca x^{2}+b x+c, where a, b and c are constants,

[ 3 ]

Question 2(ii)

(b)

the domain of f−1f^{-1}.

[ 1 ]

Question 2

[Maximum number: 3]
Question image

The diagram shows part of the curve with equation y=ksin⁡12xy=k \sin \frac{1}{2} x, where k is a positive constant and x is measured in radians. The curve has a minimum point A.

Question 2(b)

(a)

A sequence of transformations is applied to the curve in the following order.

Translation of 2 units in the negative y-direction
Reflection in the x-axis

Find the equation of the new curve and determine the coordinates of the point on the new curve corresponding to A.

[ 3 ]

Question 2

[Maximum number: 4]

A function f is defined by f(x)=x2−2x+5\mathrm{f}(x)=x^{2}-2 x+5 for x∈Rx \in \mathbb{R}. A sequence of transformations is applied in the following order to the graph of y=f(x) to give the graph of y=g(x).

Stretch parallel to the x-axis with scale factor 12\frac{1}{2}
Reflection in the y-axis
Stretch parallel to the y-axis with scale factor 3
Find g(x), giving your answer in the form ax2+bx+ca x^{2}+b x+c, where a, b and c are constants.

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